The impact of the eight most important systematic uncertainties on \(R_t\), in %.

The post-fit yields in the two SRs. All uncertainties applied in the analysis are included.

Limits on EFT parameters and Vtb extracted from the analysis. The upper and lower limits correspond to 95% CL.

Results of the main analysis.

Pre-fit distribution of the NN discriminant \(D_{\text{nn}}\) in SR plus.

Post-fit distribution of the NN discriminant \(D_{\text{nn}}\) in SR plus.

Post-fit distribution of the \(\Delta\phi(\vec{p}_{T}^{miss},\mu)\) variable in the muon-plus CR.

Pre-fit distribution of the NN discriminant \(D_{\text{nn}}\) in SR minus.

Post-fit distribution of the NN discriminant \(D_{\text{nn}}\) in SR minus.

Post-fit distribution of the \(\Delta\phi(\vec{p}_{T}^{miss},\mu)\) variable in the muon-minus CR.

Pre-fit distribution of the invariant mass of the untagged jet and the b-tagged jet, \(m(jb)\), in SR plus.

Pre-fit distribution of the absolute value of the pseudorapidity of the untagged jet, \(|\eta(j)|\), in SR plus.

Pre-fit distribution of the absolute value of the difference in \(p_{\text{T}}\) between the reconstructed W boson and the jet pair, \(|\Delta p_{\text{T}}(W, jb)|\), in SR plus.

Pre-fit distribution of the difference in azimuth angle between the reconstructed W boson and the jet pair, \(|\Delta\phi(W, jb)|\), in SR plus.

Pre-fit distribution of the invariant mass of the reconstructed top quark, \(m(t)\), in SR plus.

Pre-fit distribution of the absolute value of the difference in pseudorapidity between the charged lepton and the untagged jet, \(|\Delta\eta(\ell ,j)|\), SR plus.

Pre-fit distribution of the angular distance of the charged lepton and the untagged jet, \(\Delta R(\ell ,j)\), in SR plus.

Pre-fit distribution of the absolute value of the difference in pseudorapidity between the b-tagged jet and the charged lepton, \(|\Delta\eta(b,\ell )|\), in SR plus.

The Charged-hadron yields as a function of pT in different y selections in p+Pb collisions.

The K^{0}_{S} yields as a function of pT in different y selections in Pb+Pb photo-nuclear collisions.

The #Lambda yields as a function of pT in different y selections in Pb+Pb photo-nuclear collisions.

The #Xi^{-} yields as a function of pT in different y selections in Pb+Pb photo-nuclear collisions.

The K^{0}_{S} yields as a function of pT in different y selections in p+Pb collisions.

The #Lambda yields as a function of pT in different y selections in p+Pb collisions.

The #Xi^{-} yields as a function of pT in different y selections in p+Pb collisions.

The Charged-hadron and identified-hadron yields as a function of y in Pb+Pb photo-nuclear collisions.

The Charged-hadron and identified-hadron yields as a function of y in Pb+Pb photo-nuclear collisions.

The Charged-hadron and identified-hadron yields as a function of y in p+Pb collisions.

The Charged-hadron and identified-hadron yields as a function of y in p+Pb collisions.

The meanpT of charged hadrons and identified hadrons as a function of Nchrec in Pb+Pb photo-nuclear collisions.

The meanpT of charged hadrons and identified hadrons as a function of Nchrec in Pb+Pb photo-nuclear collisions.

The meanpT of charged hadrons and identified hadrons as a function of Nchrec in Pb+Pb photo-nuclear collisions.

The meanpT of charged hadrons and identified hadrons as a function of Nchrec in p+Pb collisions.

The meanpT of charged hadrons and identified hadrons as a function of Nchrec in p+Pb collisions.

The baryon to meson ratio as a function of pT in Pb+Pb photo-nuclear collisions.

The baryon to meson ratio as a function of pT in Pb+Pb photo-nuclear collisions.

The baryon to meson ratio as a function of pT in p+Pb collisions.

The baryon to meson ratio as a function of pT in p+Pb collisions.

The ratios of identified-strange-hadron yield to charged-hadron yields as a function of Nchrec in Pb+Pb photo-nuclear collisions.

The ratios of identified-strange-hadron yield to charged-hadron yields as a function of Nchrec in Pb+Pb photo-nuclear collisions.

The ratios of identified-strange-hadron yield to charged-hadron yields as a function of Nchrec in Pb+Pb photo-nuclear collisions.

The ratios of identified-strange-hadron yield to charged-hadron yields as a function of Nchrec in p+Pb collisions.

The ratios of identified-strange-hadron yield to charged-hadron yields as a function of Nchrec in p+Pb collisions.

Total invariant mass distribution $m_{4\ell}$ in Signal Region 2.

Selected events in the ($m_{4\ell}$, $\left\langle m_{\ell\ell}\right\rangle$) plane, SR1

Selected events in the ($m_{4\ell}$, $\left\langle m_{\ell\ell}\right\rangle$) plane, SR2

Expected background events in the $(m_{4\ell}, \langle m_{\ell\ell}\rangle)$ plane, SR1

Expected signal shape for $(m_{S}, m_{Z_d}) = (110~\text{GeV},30~\text{GeV}$) in the $(m_{4\ell}, \langle m_{\ell\ell}\rangle)$ plane, SR1. The signal distributions are normalized to unit volume.

Expected background events in the $(m_{4\ell}, \langle m_{\ell\ell}\rangle)$ plane, SR2. Events that satisfy $|m_{ab,cd} - m_Z| < 8~\text{GeV}$ are removed by the Z-veto requirement, however not all events lying behind the mask in the plot, corresponding to $\langle m_{\ell\ell}\rangle$ values between 83 and 99 GeV, are deleted.

Expected signal shape for $(m_{S}, m_{Z_{d}}) = (750~\text{GeV},150~\text{GeV}$) in the $(m_{4\ell}, \langle m_{\ell\ell}\rangle)$ plane, SR2. The signal distributions are normalized to unit volume. Events that satisfy $|m_{ab,cd} - m_Z| < 8~\text{GeV}$ are removed by the Z-veto requirement, however not all events lying behind the mask in the plot, corresponding to $\langle m_{\ell\ell}\rangle$ values between 83 and 99 GeV, are deleted.

Expected 95% CL limit on $\sigma \mathcal{B}(S\rightarrow Z_{d}Z_{d} \rightarrow 4\ell)$, SR1

Observed 95% CL limit on $\sigma \mathcal{B}(S\rightarrow Z_{d}Z_{d} \rightarrow 4\ell)$, SR1

Observed / Expected 95% CL limit ratio in SR1

Expected 95% CL limit on $\mathcal{B}(S\rightarrow Z_{d}Z_{d} \rightarrow 4\ell)$ with Z-veto, SR2

Observed 95% CL limit on $\sigma \mathcal{B}(S\rightarrow Z_{d}Z_{d} \rightarrow 4\ell)$ with Z-veto, SR2

Observed / Expected 95% CL limit ratio with Z-veto, SR2

Local $p_0$-values in the ($m_{S}, m_{Z_{d}}$) plane, SR1

Local $p_0$-values in the ($m_{S}, m_{Z_{d}}$) plane, SR2 with Z-veto applied

Observed data in Signal Region 1 in the $(m_{4\ell}, \langle m_{\ell\ell}\rangle)$ plane.

Observed data in Signal Region 2 in the $(m_{4\ell}, \langle m_{\ell\ell}\rangle)$ plane. Events that satisfy $|m_{ab,cd} - m_Z| < 8~\text{GeV}$ are removed by the Z-veto requirement, however not all events lying behind the mask in the plot, corresponding to <m_ll> values between 83 and 99 GeV, are deleted.

Distributions of the average dilepton mass $\left\langle m_{\ell\ell}\right\rangle = \frac{1}{2}\left(m_{ab} + m_{cd}\right)$ for Signal Region 1 in the $4e$ final state.

Distributions of the average dilepton mass $\left\langle m_{\ell\ell}\right\rangle = \frac{1}{2}\left(m_{ab} + m_{cd}\right)$ for Signal Region 1 in the $2e2\mu$ final state.

Distributions of the average dilepton mass $\left\langle m_{\ell\ell}\right\rangle = \frac{1}{2}\left(m_{ab} + m_{cd}\right)$ for Signal Region 1 in the $4\mu$ final state.

Distributions of the average dilepton mass $\left\langle m_{\ell\ell}\right\rangle = \frac{1}{2}\left(m_{ab} + m_{cd}\right)$ for Signal Region 2 in the $4e$ final state.

Distributions of the average dilepton mass $\left\langle m_{\ell\ell}\right\rangle = \frac{1}{2}\left(m_{ab} + m_{cd}\right)$ for Signal Region 2 in the $2e2\mu$ final state.

Distributions of the average dilepton mass $\left\langle m_{\ell\ell}\right\rangle = \frac{1}{2}\left(m_{ab} + m_{cd}\right)$ for Signal Region 2 in the $4\mu$ final state.

Distributions of the total invariant mass $m_{4\ell}$ for Signal Region 1 in the $4e$ final state. The SR1 requirement $m_{4\ell} < 115~\text{GeV}$ is not applied here.

Distributions of the total invariant mass $m_{4\ell}$ for Signal Region 1 in the $2e2\mu$ final state. The SR1 requirement $m_{4\ell} < 115~\text{GeV}$ is not applied here.

Distributions of the total invariant mass $m_{4\ell}$ for Signal Region 1 in the $4\mu$ final state. The SR1 requirement $m_{4\ell} < 115~\text{GeV}$ is not applied here.

Distributions of the total invariant mass $m_{4\ell}$ for Signal Region 2 in the $4e$ final state.

Distributions of the total invariant mass $m_{4\ell}$ for Signal Region 2 in the $2e2\mu$ final state.

Distributions of the total invariant mass $m_{4\ell}$ for Signal Region 2 in the $4\mu$ final state.

Data selection efficiency as a function of nuclear recoil energy

Data points used in analysis in log_10(S2)-S1 space

$\xi$ distributions for different jet $p_T$ bins.

$\xi$ distributions for different jet $p_T$ bins.

The $j_T/p_{T}^{\rm jet}$ distributions for different jet $p_T$ bins.

The $j_T/p_{T}^{\rm jet}$ distributions for different jet $p_T$ bins.

$r$ distributions for different jet $p_T$ bins.

Isoscalar interaction coupling limit for Lagrangian 7

Isoscalar interaction coupling limit for Lagrangian 10

Isovector interaction coupling limit for Lagrangian 14

Isoscalar interaction coupling limit for Lagrangian 6

Isovector interaction coupling limit for Lagrangian 7

Isovector interaction coupling limit for Lagrangian 9

Isoscalar interaction coupling limit for Lagrangian 18

Isoscalar interaction coupling limit for Lagrangian 4

Isoscalar interaction coupling limit for Lagrangian 2

Isovector interaction coupling limit for Lagrangian 18

Isoscalar interaction coupling limit for Lagrangian 9

Isovector interaction coupling limit for Lagrangian 1

Isovector interaction coupling limit for Lagrangian 8

Isovector interaction coupling limit for Lagrangian 5

Isovector interaction coupling limit for Lagrangian 17

Isoscalar interaction coupling limit for Lagrangian 5

Isoscalar interaction coupling limit for Lagrangian 8

Isovector interaction coupling limit for Lagrangian 12

Isoscalar interaction coupling limit for Lagrangian 11

Isovector interaction coupling limit for Lagrangian 15

Isoscalar interaction coupling limit for Lagrangian 3

Isoscalar interaction coupling limit for Lagrangian 13

Isoscalar interaction coupling limit for Lagrangian 16

Isovector interaction coupling limit for Lagrangian 3

Isovector interaction coupling limit for Lagrangian 20

Isovector interaction coupling limit for Lagrangian 4

Isoscalar interaction coupling limit for Lagrangian 14

Isoscalar interaction coupling limit for Lagrangian 12

Isovector interaction coupling limit for Lagrangian 13

Isovector interaction coupling limit for Lagrangian 11

Isoscalar interaction coupling limit for Lagrangian 20

Isoscalar interaction coupling limit for Lagrangian 17

Isoscalar interaction coupling limit for Lagrangian 15

Isovector interaction coupling limit for Lagrangian 6

Isovector interaction coupling limit for Lagrangian 2

Isovector interaction coupling limit for Lagrangian 10

Isovector interaction coupling limit for Lagrangian 16

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the combined fit, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the combined fit, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the combined fit, when all other coupling modifiers are fixed to their SM predictions.

Likelihood observed contours at 68% CL (solid line) and 95% CL (dashed line) in the ($\kappa_{\lambda}$, $\kappa_{2V}$) parameter space, when all other coupling modifiers are fixed to their SM predictions. The best-fit value, denoted by a cross in the plot, corresponds to the very first entry in the table.

Likelihood expected contours at 68% CL (teal shaded region) and 95% CL (yellow shaded region) in the ($\kappa_{\lambda}$, $\kappa_{2V}$) parameter space, when all other coupling modifiers are fixed to their SM predictions. In the plot, the SM prediction is indicated by the star.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{gghh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross. The best-fit value corresponds to the very first entry in the table.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{gghh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{tthh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross. The best-fit value corresponds to the very first entry in the table.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{tthh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{H}$, $c_{H◻}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross. The best-fit value corresponds to the very first entry in the table.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{H}$, $c_{H◻}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross.

The acceptance times efficiency [in %] for the signal ggF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel. The other coupling modifiers affecting ggF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal ggF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ SLT channel. The other coupling modifiers affecting ggF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal ggF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ LTT channel. The other coupling modifiers affecting ggF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ SLT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ LTT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{2V}$ in each analysis category for the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{2V}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ SLT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{2V}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ LTT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Summary of the expected (blue) and observed (orange) one-dimensional constraints at 68% (solid error bars) and 95% (dashed error bars) CL on the coefficients of selected HEFT (top) and SMEFT (bottom) operators describing Higgs boson interactions affecting $HH$ production through gluon-gluon fusion. The results are obtained from one dimensional scans of the profile log-likelihood assuming that all other Wilson coefficients are fixed to their SM values. The contribution from VBF $HH$ production is neglected.

Local observed significance for signal discovery at different (mX, mS). The points show where the significance was evaluated. The band at mS = 125 GeV is not shown as those points are equivalent to those already probed in Phys. Rev. D 106 (2022) 052001.

Signal region yield for each of the simulated signal samples and interpolated signal points that were analyzed in the 1 b-jet Signal Region, normalized to a cross section of 1 fb. The signal region yields for one of the signals is found in Table 2 of the paper.

Signal region yield for each of the simulated signal samples and interpolated signal points that were analyzed in the 2 b-jet Signal Region, normalized to a cross section of 1 fb. The signal region yields for one of the signals is found in Table 2 of the paper.